All drawings move together in Blender? Is there a way to turn this off?
In Grease Pencil edit mode: select the strokes you want to split from the rest and then (menu)Stoke -> Separate -> Selected Strokes. The Separate menu has the shortcut "P". You will then have the selected strokes as separate Grease Pencil objects. You can also move them in edit mode, after you select, as you see fit.